Question : The square root of $\frac{2+\sqrt{3}}{2}$ is:
Option 1: $\pm \frac{1}{\sqrt{2}}(\sqrt{3}+1)$
Option 2: $\pm \frac{1}{2}(\sqrt{3}-2)$
Option 3: $\text{none}$
Option 4: $\pm \frac{1}{2}(\sqrt{3}+1)$
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Correct Answer: $\pm \frac{1}{2}(\sqrt{3}+1)$
Solution :
Given: $\frac{2+\sqrt{3}}{2}$
$= \frac{1}{4}({4+2\sqrt{3}})$
$= \frac{1}{4}[1^2+(\sqrt{3})^2+2×1×\sqrt{3}]$
$= \frac{1}{4}(1+\sqrt{3})^2$
So, the square root of $\frac{2+\sqrt{3}}{2}$
$= \sqrt{\frac{2+\sqrt{3}}{2}}$
$= \sqrt{\frac{1}{4}(1+\sqrt{3})^2}$
$= \pm \frac{1}{2}(1+\sqrt{3})$
Hence, the correct answer is $\pm \frac{1}{2}(\sqrt{3}+1)$ .
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